I'm a bit confused on the countable chain condition, as I have encountered two different definitions of the concept, and I am not sure if they are necessarily equivalent.
When talking about partial orders, one says that a partially ordered set $P$ satisfies ccc if every strong antichain in $P$ is countable.
When talking about total orders, one says that a totally ordered set $T$ satisfies ccc if every pairwise disjoint collection of open subsets of $T$ is countable, where a subset is open if it is the union of open intervals.
My confusion is that since every element of a total order is comparable, a total order must trivially satisfy the first notion of ccc since there are no antichains to begin with. But because the second definition exists, it suggests that there are totally ordered sets which don't satisfy the second notion of ccc.
So in short, my question is asking if these two definitions of ccc are equivalent in some way I am not understanding.
I write "1-c.c.c." and "2-c.c.c." for the two definitions of c.c.c.
The two definitions are not directly equivalent - there are total orders which are not 2-c.c.c. (consider e.g. the first uncountable ordinal $\omega_1$), and as you observe any total order is trivially 1-c.c.c.
However, there is a relationship: to a total order $L$ we let $Int(L)$ be the interval order; this is the partial order whose elements are open intervals of $L$, with the order relation being $\subseteq$. Note that $Int(L)$ is a partial order: for instance, if $L=(\mathbb{R}, <)$ then $(0, 1)$ and $(2, 3)$ are incompatible elements of $Int(L)$. Then we have